```From: glhurst@onr.com (Gerald L. Hurst)
Newsgroups: sci.physics
Subject: Re: A New Gas Pressure Theory (Feb29)
Date: 2 Mar 1996 01:01:05 GMT

Blakely <74213.2646@CompuServe.COM> says:

>                      A New Theory of Gas Pressure
>
>Revised February 29, 1996, by Richard Blakely
>The purpose of this file: to expose defects in quantum theory as it is
>applied to gases.
>
>THE OLD GAS PRESSURE THEORY:
>     Kinetic-molecular theory was developed as a result of the work of
>Boltzmann, Maxwell, and Clausius at the end of the 1800s.  The pressure
>which a molecule exerts on a container surface is due to its momentum
>change.  A molecule exerts an impulse of m*2*v on a surface when it
>collides and then rebounds.  A large number of molecules in a cube shaped
>box one unit across exert a pressure P=(N/V)*m*v*v where v is the
>perpendicular velocity, assuming that all the molecules move
>perpendicular to one side of the box.  Since real molecules do not move
>only in a direction perpendicular to a box surface, a factor of 1/2 is
>required to reduce pressure.  Pressure must be reduced by 1/2 because the
>surface area of a half-sphere is twice the surface area of a circle of
>the same diameter.  Including 1/2 in the equation for pressure of a gas
>it becomes P=(N/V)*m*v*v/2, or P=(n/V)*M*v*v/2 using mole mass.  Since
>PV=nRT=n*M*v*v/2, the kinetic energy of a mole of gas molecules might
>seem to be M*v*v/2.  As an example of what this means, the rms speed of
>an argon molecule at one atmosphere and 273 degrees might be calculated
>as the square root of (2*R*T/M) which is 337 m/s, since M=.03995 and
>R=8.3143.
>     Actually, the energy (temperature x specific heat) of a argon gas is
>(3/2)RT, which means that the rms speed of a typical argon molecule is
>(3*R*T/M)^.5 = 413 m/s, not 337 m/s.  Quantum gaseous theory, as
>described in physics books, states that the factor of 3/2 arises because
>of the "fantasy" that velocity can be clumped into only three directions
>x, y, and z.  The theory is that for each of the three dimensions a
>factor of 1/2 is required, and thus 3/2 results because of the three
>dimensions.  But is it possible that molecules can only move in exactly
>three directions, and if so then which ones, of the infinite possible
>directions, are those three?  Are we justified in believing the "quantum-
>fantasy" because every physics book restates the faulty theory?  If we
>are, then we should also believe in a flat earth because it appears to be
>flat!

The 3/2 is not the result of some quantum fantasy. Rather it arises
from a simple classical analogy based on the fact that all collisions
produce momentum changes which can be resolved into 3 components
directed along  three arbitrary perpendicular reference axes X,Y,Z.
If you imagine a frictionless ball of mass m bouncing around in
a cubical box of dimension L^3 = V then the rate of change of momentum
is 2*m*v*v/L, a force which is distributed over an area of 6*L^2,
which we divide into the force to get the pressure:

P = 2/6*m*v*v/L^3 which is the same as PV = 1/3*mv^2

If we increase the number of balls to N the impact frequency and
thus the pressure increases proportionately

PV = 1/3Nmv^2

Now, if N is the number of atoms and m is the unit atom weight, we
can rewrite this equation in  numerically equal terms of moles (n)
and mole weight (M):

PV = 1/3nMv^2 = nRT (we know the nRT from experiments)

Solving for velocity we get:

v = sqrt(3RT/M) without invoking any quantum ideas.

Let's back up now and look at the validity of  the original
assumption that led us to that critical number "3."  The
easiest way is to imagine a box with cubic dimensions L in which
a frictionless marble of mass m is bouncing from end to end hitting
only two opposing walls. The average force exerted is equal to
the momentum change per unit time which is the impact frequency
(v/L) multiplied by the momentum change (2*m*v) which yields
2mv^2/L. The question is whether we are now justified in saying
that we can estimate the AVERAGE force on all the walls from that
same marble now bouncing randomly against all the walls by simply
dividing by three. It turns out that we can reasonably confidently
do so based on cut-and-try calculations in which we bounce a
hypothetical marble off the walls like a billiard ball making
three dimensional rail shots. For instance, if we let the marble
go round and around four walls on a plane parallel to the untouched
walls we soon find that regardless of the angle the total rate
of momentum change is constant for constant velocity, but that
change is occurring over twice as many walls so the force per
wall is reduced to half. Intuitively, one can see that extending
the bounce to three dimensions will drop the force per wall to
one third of the original value for a normal 2-wall bounce. That
intuition can be confirmed by a few calculated examples.

If you wish, you can replace the single marble with two marbles
with unchanged total momentum to show that the momentum change
rate (and thus the pressure) remains unchanged and thereby
prove to yourself that extrapolation to large numbers of smaller
marbles (molecules) is valid.

Obviously, a mathematician would quickly reduce my cut-an-try
explanation to a simple mathematical proof, but spit-and-bailing-
wire science sometimes likes a little assist from simple thought
experiments.

>A NEW GAS PRESSURE THEORY:
>     Assuming the "quantum fantasy" is faulty, the fault is due to one of
>three possibilities:  It might be that some gas molecules move more
>slowly than assumed, since in the equation PV=nRT=n*M*v*v/2, v seems
>slower than 413m/s.  Or, a second possibility is that (n/V) is less than
>specified in the equation, and then v=413m/s would be true.  But (n/V)
>must be correct because gases have been weighed.  A third possibility,
>and the most likely cause of the defect, is that the original "quantum-
>fantasy" assumption has a logical error.  That logical error is similarly
>used in other quantum theories, for example, in electron quantum theory.
>     The logical error in the "quantum-fantasy" is the use of a cube to
>hold a molecule of gas.  Gas molecules do not confine themselves in small
>cubes.  A more realistic model of gas molecule motion is to use a
>spherical container, instead of a cube, to hold a single molecule.  In a
>sphere, as in a cube, a molecule can move about randomly, but the average
>molecular-motion path length is more realistic in a sphere because a real
>molecule moves about in Brownian motion, which means that it maintains
>its approximate average position in the center of its "sphere of motion".
>Sphere volume can be calculated as (pi/6)d^3 where d is sphere diameter.
>Average path length in a sphere can be found using several equations to
>be p=d*cos(35.26439 degrees)=d*.816497 where p is path length.
>Converting to path length per unit volume, path length becomes
>p*6/pi=1.5594.  Using p in the equation previously developed, the
>equation becomes P=((n/V)/p)*M*v*v/2.  The value of 1/p is close to the
>"ideal" 2/3 where p is average path length per unit volume length.  Thus,
>the modified equation can use the rms speed v as set by (3/2)RT or
>(3*R*T/M)^.5 = 413 m/s.

Whoa, by your own premises the length per unit volume is not
1.5594 but must be 1.5594/d^2. You cannot normalize d to a value
of 1 because it is a function of V which you are using in your
overall equation.  You have trod on the slippery slope looking
for your 2/3 figure, i.e., the magic "3" discussed above.

The problem is not one of "cube versus sphere" assumptions. The
problem is simply more difficult to work with as a thought
experiment using a sphere for the simple reason that we must
always account for momentum components in three dimensions, and
the projections of a sphere onto three perpendicular dimensional
planes is a bit more difficult than with a cube whose sides are
in effect their own projections.

Whether we speak of cubes or spheres is irrelevant to the theory.
These are mere thought tools with neither having any fundamental
significance.  It appears to me at this point that you are
trying to mend a theory that is not broken.

Jerry (Ico)
```